Lost my last post, will try again.
First off, does the phone come come up ON when you plug it into power? If so it would be a lot easier to turn a relay ON for 10 minutes and drop it out for 5 seconds then back on.
Also, if you put a relay across the power button with what you describe, you will need the phone on to start the process to keep it sync'd like you want.
I would use a PIC and Atmel chip, but now that people know what you are wanting to do, sure other will offer a discrete or logic way to do it.